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Nonlinear equations in one variable and systems of equations in two variables SAT question

Math · Advanced Math · Hard · Answer it here, explanation included.

In the xy-plane, the graph of $y equals 3 x squared minus 14 x$ intersects the graph of $y equals x$ at the points $with coordinates zero comma zero$ and $a, comma a$ . What is the value of a ?

Enter a number or fraction (like 3/17 or 0.176). Press Enter to check.

ExplanationShow

The correct answer is 5. The intersection points of the graphs of $y equals, 3 x squared, minus 14 x$ and $y equals x$ can be found by solving the system consisting of these two equations. To solve the system, substitute x for y in the first equation. This gives $x equals, 3 x squared, minus 14 x$ . Subtracting x from both sides of the equation gives $0 equals, 3 x squared, minus 15 x$ . Factoring $3 x$ out of each term on the left-hand side of the equation gives $0 equals, 3 x times, open parenthesis, x minus 5, close parenthesis$ . Therefore, the possible values for x are 0 and 5. Since $y equals x$ , the two intersection points are $the points with coordinates 0 comma 0$ and $5 comma 5$ . Therefore, $a, equals 5$ .

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Nonlinear equations in one variable and systems of equations in two variables SAT question

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